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# Doodling in math: Snakes andย graphs

## Video transcript

So you're me and
you're in math class, and you're learning
about graph theory. A subject too interesting to be
included in most grade school curricula, so maybe you're
in some special program, or maybe you're in
college and somehow not scarred for life by your
grade school math teachers. I'm not sure why you're
not paying attention, but maybe you have an
incompetent teacher, and it's too heartbreaking
to watch him butcher what could have been such a
fun subject full of snakes and balloons. Snakes aren't really all that
relevant to the mathematics here, but being able to draw
them will be useful later. So you should probably
start practicing now. I've got a family of
three related doodle games to show you, all stemming
from drawing squiggles all over the page. The first one goes like this. Draw a squiggle. A closed curve that
ends where it begins. The only real rule
here is to make sure all the crossings
are distinct. Next, make it start weaving. Follow the curve around, and at
each crossing, alternate going under and over until you've
assigned all the crossing. Then put on the finishing
touches, and voila. You try it again, adding
a little artistic flair to the lines. The cool part is that
the meeting always works out perfectly. When you're going around
alternating over and under, and get to a crossing
you've already assigned, it will always be the right one. This is very interesting, and
we'll get back to it later. But first I'd like to
point out two things. One is that this works for
any number of closed curves on the plane, so go ahead
and link stuff up or make a weaving out of
two colors of yarn. The other is that
this doodle also works for snakes on a plane,
as long as you keep the head and tail on the outside or
on the same inside face, because mathematically it's
the same as if they linked up. Or just actually link
up the head and tail, into an Ouroboros. For example, here's
three Ouroborii in a configuration known as
the Borromean Rings, which has the neat property that
no two snakes are actually linked with each other. Also, because I
like naming things, this design shall henceforth
be known as the Ouroborromean Rings. But you are me,
after all, so you're finding a lot to think
about, even with just drawing one line that isn't a snake. Such as what kind of
knot are you drawing? And can you classify them? For example, these three
knots all have five crossings, but two are essentially the
same knot, and one is different. Knot theory questions
are actually really difficult
and interesting, but you're going to have
look that one up yourself. Oh, and you should also
learn how to draw rope, because it's an integral
part of knot theory. So integral in fact,
that if you draw a bunch of integral
signs in a row-- a sight which is often quite
daunting to a mathematician-- you can just shade
it in, and tada. But being able to draw
snakes is also super useful, especially as this
doodle game is excellent for producing
Dark Mark tattoo designs. Also, this doodle game can be
combined with the stars doodle game. For example, if this
pentagram gets knighted, it will henceforth be
known as Sir Pentagram. Also notice that this snakes
is a five twist Mobius strip, so you could also
call it a Mobiaborros, but we'll get back to
one sidedness later. Or if you want to draw
something super complicated, like the eighth square star,
combining snakes and stars is a great technique
for that to. Here is a boa that
ate eight eight-gons. The creativity that
your mind is forced into during these boring classes
is both a gift and a burden, but here's a few authentic
doodles using these techniques that I did when I
was in college, just to show you that I'm
not making all this up. These are from a freshman
music history class, because I happen to be able
to find this notebook-- but this is a doodle I
actually did most often during my ninth grade
Italian class-- language being another subject
usually taught by unfathomably stupid methods. For example, these snakes are
having trouble communicating, because one speaks
in parseltongue, and the other speaks in Python. And their language classes,
much like math classes, focus too much on memorization,
and not enough on immersion. But just pretend
you're in math class, learning about graph theory,
so I can draw the parallels. Because here's the
second doodle game, which is very much
mathematically related. Draw a squiggle
all over the page and make sure it closes up. Pick an outside section
and color it in. Now you want to
alternate coloring so that no two faces of
the same color touch. Curiously enough, much
like the weaving game, this game also always
mathematically works out. It also works really well
if you make the lines spiky instead of a smooth curve. And once again, it works
with multiple lines too. It probably has something to
do with the two-colorability of graphs of even
degree, which might even be what your teacher
is trying to teach you at this very moment, for
all you're paying attention. But maybe you can chat with
him after class about snakes and he'll explain it to
you, because I'd rather move on to the next doodle game. This is a combination
of the last two. Step one, draw a
smooth, closed curve. Step two assign
overs and unders. Step three, shade
in every other face. After that, it takes a
little artistic finesse to get the shading
right, but you end up with some sort of
really neat surface. For example, this one has
only one edge and one side. But if you're
interested in this, you should really be talking
to your resident topology professor, and not me. But here's the thing, if someone
asked you five minutes ago, what tangled up snakes,
demented checkerboards, and crazy, twisty
surfaces have in common, what would you have answered? This is why I love mathematics. The moment when you realize
that something seemingly arbitrary and confusing is
actually part of something. It's better than the
cleverest possible ending to any crime
show or mystery novel, because that's
only the beginning. Anyway, have fun with that.